Singularity avoidance for five-axis machine tools through introducing geometrical constraints

Singularities in five-axis machining are a series of deadly positions, which may cause unstable axis movements and deteriorate the performance of machine tools. This paper presents an optimization strategy to locally deform the tool path so that singularities of five-axis machine tools can be avoided. To achieve this purpose, a new concept of “forbidden circle”, which is the projection of the three dimensional “singular cone” associated with tool orientations in a two dimensional space named P-plane, is established for the first time to transform the complicated three dimensional problem into a two dimensional problem. Based on this idea, the projected points in P-plane are interpolated as B-spline first, and then, the B-spline control points are locally optimized by taking the “forbidden circle” as the geometrical constraint to keep the B-spline from crossing it. Especially, in the optimization procedure, the constraints are linearized with the changes of B-spline control points to change the whole optimization problem as a typical positive definite quadratic programming problem, which can achieve one and only global optimal solution. By doing so, singularity is avoided, and at the same time, the machining errors caused by tool path deformation can be minimized. Simulation and experimental results verify the effectiveness of the singularity avoidance method.